Actually information message shown not than end of document reached, but when start of search reached (position from which user has started the search). And this is relatively complex to achieve for F3 search (Repeat Search) because it is very complex to find out what is
start of search in this case (if user moves cursor, if he edits something, if he opens the find dialog and changes something, but have not executed search etc). For Quick Search Bar and Find Dialog it is rather easy: start is when user sets focus to dialog or bar.
So the only feasible function here can be notification about wrap of search (depends on search direction). But small problem here, that this is already not in sync with interactive functionality. So the decision here is not obvious. I will mark the request in todo and will think further.
About the Beep at the end. This can be added also (as it is now for Quick Search Bar), but
it can be annoying for other users.